Cyclic Meir-Keeler Contraction and Its Fractals

Abstract

In present times, there has been a substantial endeavor to generalize the classical notion of iterated function system (IFS). We introduce a new type of non-linear contraction namely cyclic Meir-Keeler contraction, which is a generalization of the famous Banach contraction. We show the existence and uniqueness of the fixed point for the cyclic Meir-Keeler contraction. Using this result, we propose the cyclic Meir-Keeler IFS in the literature for construction of fractals. Furthermore, we extend the theory of countable IFS and generalized IFS by using these cyclic Meir-Keeler contraction maps.

Keywords:

• Cyclic Meir-Keeler contraction
• fixed point
• Fractal
• Fractal interpolation function
• iterated function system

AMS KEYWORDS:

• 28A80
• 37C25
• 37C70
• 47H04
• 47H10
• 54H25

Acknowledgments

The authors are grateful to Dr. P. Veeramani for his suggestion to use cyclic contraction in IFS theory. The second author is grateful to the University of Zaragoza for facilitating a visit during July, 2019.

Additional information

Funding

This article endorses a partial financial support from the MATRICS project MTR/2017/000574 of the Science and Engineering Research Board, Government of India.